Abelian Functions and Number Theory

Overview

The theory of elliptic functions has a long history and is closely related to many areas of mathematics, including number theory, algebraic geometry, differential equations, and integrable systems. This workshop will focus on the theory of abelian functions, which is a generalization of elliptic functions. Topics will include theta functions, sigma functions, and higher-dimensional Jacobian varieties. We will discuss both theoretical and experimental aspects as well as applications to number-theoretic problems. We welcome participants from a wide range of related fields.

Abstracts

• Julia Bernatska (University of Connecticut) (online)

  • Title: Multiply periodic functions as solutions to integrable hierarchies

  • Abstract: As the main subject, we consider multiply periodic functions, also known as Kleinian $\wp$-functions, which generalize the Weierstrass $\wp$-function to higher genera. Associated with a curve in the Weierstrass canonical form, these Abelian functions form a differential field that serves as a powerful tool for uniformizing the curve. This field possesses numerous identities for $\wp$-functions, including cubic relations defining the corresponding Jacobi variety, dynamical flows known as Hirota equations, and addition laws.

The $\wp$-functions also play a significant role in the theory of integrable hierarchies associated with soliton-type equations. Integrable hierarchies admit algebraic integration on the corresponding spectral curves, which produces solutions expressed in terms of functions that uniformize these curves. Solutions to some well-known completely integrable equations will be presented, including the KdV, MKdV, sine(sinh)-Gordon, Boussinesq equations, and some equations constructed as BKM systems, namely the Kaup---Boussinesq and Camassa---Holm equations. These solutions have the advantage of being global, which allows for revealing reality conditions and analyzing the connection between the shape of a solution and the parameters of the spectral curve. The cases of soliton solutions and solitons on a wave background arise when the spectral curve degenerates to lower genera, and explicit solutions can be obtained.

• Tetsushi Ito (Kyoto University)

  • Title: TBA

  • Abstract: TBA

• Yaacov Kopeliovich (University of Connecticut) (online)

  • Title: The second Thomae formula for cyclic covers of order 3 and Schottky type relations for them

  • Abstract: Nakayashiki in tour de force showed the Thomae formula for cyclic covers of CP^{1}. In this talk I will outline the proof of the second Thomae formula that evaluates the derivative of theta functions evaluated at certain divisors for certain cyclic covers of order 3. In the second part of the talk I will show how Nakayashiki Thomae formula leads to Schottky type relations involving products of roots of theta functions for cyclic covers of order 3. 

The first part of this talk is a joint work with Shaul Zemel and the late Enolskii whose ideas in the hyperelliptic case led to the generalization discussed above.

• Toshiki Matsusaka (Kyushu University)

  • Title: Applications of indefinite theta functions

  • Abstract: In contrast to the positive definite case, theta series associated with indefinite quadratic forms involve a tension between convergence and modularity. Various approaches have been developed to overcome this difficulty, including Zwegers’ theory of indefinite theta functions. Originally introduced in his study of Ramanujan’s mock theta functions, this theory has since found applications in several different areas. In this talk, I will introduce some of these applications, mainly from number theory and related topics.

• Atsushi Nakayashiki (Tsuda University)

  • Title: Addition formulas of theta functions and integrable systems

  • Abstract: Theta functions of Riemann surfaces satisfy a famous addition formula called Fay's determinant identity. Fay has derived two limits of it. They correspond to the Frobenius-Stickelberger formulas for the Weierstrass sigma function. In the study of integrable systems we have found another limit formula of Fay's identity. In this talk we will discuss it.

• Ryo Ohashi (University of Tokyo)

  • Title: Computing isogeny graphs via theta functions

  • Abstract: In this talk, we primarily outline an algorithm based on theta functions for computing (2,2,2)-isogeny graphs of principally polarized abelian threefolds. This algorithm has applications to counting superspecial hyperelliptic curves of genus 3 and to constructing cryptographic hash functions believed to be resistant to quantum attacks. We also discuss future directions and the challenges involved in extending these results to higher dimensions.

• Yoshihiro Ônishi (Meijo University)

  • Title: On some generalizations of number-theoretic aspects of elliptic functions to Abelian functions

  • Abstract: Almost all deep theories in algebraic numbers are encapsulated in zeta functions. While zeta functions are often reluctant to reveal their nature, elliptic functions (like trigonometric functions) elucidate them and clarify their number-theoretic properties. Indeed, many examples demonstrate how elliptic functions serve as a bridge to unveil these properties. Driven by these ideas, I have been working on Abelian functions to elevate their theory toward the sophistication of elliptic functions. In this presentation, I will review several results obtained so far with collaborators, including our most recent result, comparing each issue with its counterpart in the theory of elliptic functions.

Past related workshop

• Workshop on Number Theory and Integrable Systems, June 2-4, 2025, Kobe University